Problem: Let $h(x)=-3x^3+5x-2$. Find $h'(3)$. Choose 1 answer: Choose 1 answer: (Choice A) A $-22$ (Choice B) B $32$ (Choice C) C $-76$ (Choice D) D $-68$
Solution: Let's first find the expression for $h'(x)$ and then evaluate it at $x=3$. According to the sum rule, the derivative of $-3x^3+5x-2$ is the sum of the derivatives of $-3x^3$, $5x$, and $-2$. The derivatives of these terms can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ For example, this is the derivative of the first term: $\begin{aligned}\dfrac{d}{dx}(-3x^3)&=-3\dfrac{d}{dx}(x^3)&&\gray{\text{Constant multiple rule}}\\\\ &=-3\cdot (3x^2)&&\gray{\text{Power rule}}\\ \\ &=-9x^2\end{aligned}$ Here is the complete differentiation process: $\begin{aligned} &\phantom{=}h'(x) \\\\ &=\dfrac{d}{dx}(-3x^3+5x-2) \\\\ &=-3\dfrac{d}{dx}(x^3)+5\dfrac{d}{dx}(x)-\dfrac{d}{dx}(2)&&\gray{\text{Basic differentiation rules}} \\\\ &=-3\dfrac{d}{dx}(x^3)+5\dfrac{d}{dx}(x)-0&&\gray{\text{Constant rule}} \\\\ &=-3\cdot 3x^2+5\cdot 1x^0&&\gray{\text{The power rule}} \\\\ &=-9x^2+5 \end{aligned}$ So we found that $h'(x)=-9x^2+5$. Plugging in $x=3$ and evaluating using the calculator, we find that $h'(3)=-76$. In conclusion, $h'(3)=-76$.